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Full text of "Mathematical And Physical Papers - Iii"

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in so far as the pressure, and consequently the temperature, are affected by small vibratory movements. Let the fluid be referred to the rectangular axes of #, y, z ; let u9 vy w be the components of the velocity, t the time, p the pressure, p the density in equilibrium, p (1 + s) the actual density, so that s is the condensation. The three ordinary equations of motion and the equation of continuity become in this case, on neglecting as usual the squares of small quantities,
dp __       du    dp __      dv    dp          dw             ,_ ,
dx~~~p^t' ty = ~pdi} ~fa~~pdi ......... (1)}
ds_    du    dv    dw _ n                                             f>.
dt    dx    dy    dz ~~      ................ ............... ^
Let #0 be the temperature in equilibrium, #0 + 6 the actual temperature. Then p = k0p (1 + s) (1 + a0#0 + 6). Putting k for kQ(l + a0#0), a for 0(1 + 000)~l, and neglecting the product of s and 0, which are both small quantities of the first order, we get
s + cL0)   ..................... (3).
It remains to form the equation relating to the changes of temperature. Let fis be the elevation of temperature produced by a sudden small condensation s. The condensation which a given element of the fluid receives in the time dt is equal to sdt, where
,     ds       ds       ds        ds    ds          ,
s = -j- 4- u y~ +v-j-+w-7  = -T: , nearly : dt       dx      dy       dz    dt           J
and the elevation of temperature due to this condensation is equal to fts'dt. We know that heat radiates freely to great distances in air, and therefore, of the heat which radiates from the element considered, we may neglect the small portion which may be absorbed by the air in its neighbourhood, and consider only what goes to great distances. Hence the result will be sensibly the same a if the element radiated into a medium having the constant temperature 00, which is the mean temperature of the whole. The quantity, then, which escapes from the element during the time dt, will be proportional to the small excess 6 of the temperature of the element over the mean temperature of the medium ; and the consequent depression of temperature may be expressed by qOdtj where q is a constant which may be called the velocity of